Group #36469099 — subgroups

Subgroups — #36469099

Order 22784.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022502	C₂	1	1	yes	maximal, characteristic
2	#26022477	C₂	1	2	yes	maximal, complemented
3	#26019845	C₂	1	2	yes	maximal, complemented
4	#26018168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2254	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₂₄	—	2	2	no	—
7	#19151975	C₂²	1	2	yes	—
8	#19151972	—	2	2	no	—
9	#19151946	—	2	2	no	—
10	#19151398	—	2	4	no	—
11	C₄×#13083549	C₂²	1	1	yes	characteristic
12	C₈×#10623494	C₂²	1	1	yes	characteristic
13	#500×C₈₉	C₂²	1	2	yes	—
14	#367×C₈₉	—	2	2	no	—
15	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#13083561	—	4	4	no	—
17	C₂×D₇₁₂	—	2	2	no	—
18	C₂×D₇₁₂	—	2	2	no	—
19	#15668009	—	2	2	no	—
20	D₁₄₂₄	—	4	4	no	—
21	C₂×#13083549	D₄	1	2	yes	—
22	#15667932	—	4	4	no	—
23	C₄×#10623494	D₄	1	2	yes	—
24	C₄⋊Q₈×C₈₉	—	2	2	no	—
25	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
26	#107×C₈₉	—	2	2	no	—
27	#107×C₈₉	—	2	2	no	—
28	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×D₃₅₆	—	4	2	no	—
30	C₂×D₃₅₆	—	4	2	no	—
31	C₂×D₃₅₆	—	4	2	no	—
32	#13083561	—	8	8	no	—
33	D₇₁₂	—	4	4	no	—
34	D₇₁₂	—	4	4	no	—
35	C₂×#10623494	D₈	1	4	yes	—
36	#13083549	—	2	2	no	—
37	C₂×Q₈×C₈₉	—	4	2	no	—
38	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₈₉	—	4	2	no	—
41	C₄⋊C₄×C₈₉	—	4	2	no	—
42	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₈₉	SD₁₆	1	4	yes	complemented
44	C₂²×D₈₉	—	8	2	no	—
45	Q₈×C₈₉	—	8	4	no	—
46	C₂×C₄×C₈₉	—	8	2	no	—
47	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
49	D₃₅₆	—	8	2	no	—
50	D₃₅₆	—	8	2	no	—
51	D₃₅₆	—	8	4	no	—
52	D₃₅₆	—	8	4	no	—
53	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
54	C₈×C₈₉	#135	1	2	yes	—
55	#10623494	—	2	4	no	—
56	C₂²×C₈₉	#493	1	1	yes	characteristic
57	C₂×D₈₉	—	16	2	no	—
58	C₂×D₈₉	—	16	2	no	—
59	C₂×D₈₉	—	16	2	no	—
60	C₄×C₈₉	—	16	2	no	—
61	C₄×C₈₉	#492	1	1	yes	characteristic
62	C₄×C₈₉	#495	1	1	yes	characteristic
63	C₄×C₈₉	#509	1	4	yes	—
64	#12506	—	89	1	no	maximal, Sylow
65	C₂×C₈₉	#1706	1	1	yes	characteristic
66	C₂×C₈₉	#2250	1	2	yes	—
67	D₈₉	—	32	2	no	—
68	#2254	D₈₉	1	1	yes	characteristic, complemented
69	#2246	—	89	1	no	—
70	#2240	—	89	2	no	—
71	#1706	—	89	2	no	—
72	#1311	—	89	1	no	—
73	C₈₉	#12506	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₁₆	—	178	2	no	—
75	#500	C₂×D₈₉	1	2	yes	—
76	#493	—	89	2	no	—
77	#492	—	178	2	no	—
78	#486	—	178	2	no	—
79	#367	—	2	2	no	—
80	#357	—	178	4	no	—
81	C₄×C₁₆	—	89	1	no	—
82	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
83	C₈²	—	89	1	no	—
84	C₂×SD₁₆	—	356	4	no	—
85	C₂×D₈	—	178	2	no	—
86	C₂×D₈	—	178	2	no	—
87	C₄⋊Q₈	—	2	2	no	—
88	#127	—	178	2	no	—
89	D₁₆	—	356	4	no	—
90	C₂×C₁₆	—	89	2	no	—
91	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
92	#107	—	2	2	no	—
93	#107	—	2	2	no	—
94	#102	—	356	4	no	—
95	C₄×C₈	—	89	2	no	—
96	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
97	C₂×Q₈	—	4	2	no	—
98	C₂×D₄	—	356	2	no	—
99	C₂×D₄	—	356	2	no	—
100	C₂×D₄	—	356	2	no	—
101	SD₁₆	—	712	8	no	—
102	D₈	—	356	4	no	—
103	D₈	—	356	4	no	—
104	C₂×C₈	—	89	4	no	—
105	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄²	D₄×D₈₉	1	1	yes	characteristic
110	C₁₆	—	178	2	no	—
111	C₁₆	#13083561	1	4	yes	complemented
112	C₂³	—	712	2	no	—
113	Q₈	—	8	4	no	—
114	D₄	—	712	2	no	—
115	D₄	—	712	2	no	—
116	D₄	—	712	4	no	—
117	D₄	—	712	4	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	#15668008	1	1	yes	characteristic
120	C₂×C₄	D₈×D₈₉	1	2	yes	—
121	C₈	—	178	4	no	—
122	C₈	C₂×#13083561	1	2	yes	—
123	C₈	#15668035	1	2	yes	—
124	C₂²	—	1424	2	no	—
125	C₂²	—	1424	2	no	—
126	C₂²	—	1424	2	no	—
127	C₂²	#19151973	1	1	yes	characteristic
128	C₄	—	16	2	no	—
129	C₄	#19151972	1	1	yes	characteristic
130	C₄	#19151983	1	1	yes	characteristic
131	C₄	#19152035	1	4	yes	—
132	C₂	—	2848	2	no	—
133	C₂	#26019845	1	1	yes	characteristic
134	C₂	#26022517	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022502

C₂

yes

maximal, characteristic

#26022477

C₂

yes

maximal, complemented

#26019845

C₂

yes

maximal, complemented

#26018168

C₂

yes

maximal, characteristic, complemented

C₈₉×#2254

C₂

yes

maximal, characteristic, complemented

C₂×D₁₄₂₄

—

#19151975

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₈×#10623494

C₂²

yes

characteristic

#500×C₈₉

C₂²

yes

—

#367×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×#13083561

—

C₂×D₇₁₂

—

C₂×D₇₁₂

—

#15668009

—

D₁₄₂₄

—

C₂×#13083549

D₄

yes

—

#15667932

—

C₄×#10623494

D₄

yes

—

C₄⋊Q₈×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

—

#107×C₈₉

—

#107×C₈₉

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

#13083561

—

D₇₁₂

—

D₇₁₂

—

C₂×#10623494

D₈

yes

—

#13083549

—

C₂×Q₈×C₈₉

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

SD₁₆

yes

complemented

C₂²×D₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₈×C₈₉

C₂×SD₁₆

yes

—

C₈×C₈₉

#135

yes

—

#10623494

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

#2254

D₈₉

yes

characteristic, complemented

—

—

—

—

C₈₉

#12506

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

178

—

#500

C₂×D₈₉

yes

—

#493

—

#492

—

178

—

#486

—

178

—

#367

—

#357

—

178

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₉

yes

characteristic

C₈²

—

C₂×SD₁₆

—

356

—

C₂×D₈

—

178

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

#127

—

178

—

D₁₆

—

356

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₈₉

yes

—

#107

—

#107

—

#102

—

356

—

C₄×C₈

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

100

C₂×D₄

—

356

—

101

SD₁₆

—

712

—

102

D₈

—

356

—

103

D₈

—

356

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₈₉

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₈₉

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

D₄×D₈₉

yes

characteristic

110

C₁₆

—

178

—

111

C₁₆

#13083561

yes

complemented

112

C₂³

—

712

—

113

Q₈

—

114

D₄

—

712

—

115

D₄

—

712

—

116

D₄

—

712

—

117

D₄

—

712

—

118

C₂×C₄

—

119

C₂×C₄

#15668008

yes

characteristic

120

C₂×C₄

D₈×D₈₉

yes

—

121

C₈

—

178

—

122

C₈

C₂×#13083561

yes

—

123

C₈

#15668035

yes

—

124

C₂²

—

1424

—

125

C₂²

—

1424

—

126

C₂²

—

1424

—

127

C₂²

#19151973

yes

characteristic

128

C₄

—

129

C₄

#19151972

yes

characteristic

130

C₄

#19151983

yes

characteristic

131

C₄

#19152035

yes

—

132

C₂

—

2848

—

133

C₂

#26019845

yes

characteristic

134

C₂

#26022517

yes

—

	subgroup	G/H
center	C₂²	#19151973
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#2254	C₂
socle	C₂²×C₈₉	#493
prime radical	#26022502	C₂
hypercenter	#2254	D₈₉
hypocenter	C₈₉	#12506
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#19151973

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#2254

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
2. #26022477	C₂	semidirect
3. #26019845	C₂	semidirect
4. #26018168	C₂	semidirect
5. C₈₉×#2254	C₂	semidirect
43. C₁₆×C₈₉	SD₁₆	semidirect
68. #2254	D₈₉	semidirect
73. C₈₉	#12506	semidirect
111. C₁₆	#13083561	semidirect

H ≅ G/N

type

2. #26022477

C₂

semidirect

3. #26019845

C₂

semidirect

4. #26018168

C₂

semidirect

5. C₈₉×#2254

C₂

semidirect

43. C₁₆×C₈₉

SD₁₆

semidirect

68. #2254

D₈₉

semidirect

73. C₈₉

#12506

semidirect

111. C₁₆

#13083561

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26022502	2
#26022477	2
#26019845	2
#26018168	2
C₈₉×#2254	2
#12506	89

subgroup

index

C₈₉×#2254

Subgroups — #36469099

Conjugacy classes of subgroups

Complements

Maximal subgroups